Q. 1. The values of f(x) lie in the interval
Ans.
Solution. For the given function to be defined
and sine function increases on [0, π/4]
Q. 2. For the function
the derivative from the right, f '(0+) = ......................., and the derivative from the left, f '(0–) = ...............
Ans. 0, 1
Solution.
Thus f ' (0^{+}) = 0 and f ' (0^{–}) = 1
Q. 3. The domain of the function given by .................
Ans. [–2, –1] ∪ [1, 2]
Solution. To find domain of function
For f (x) to be defined we should have
NOTE THIS STEP :
Q. 4. Let A be a set of n distinct elements. Then the total number of distinct functions from A to A is ................. and out of these ................. are onto functions.
Ans. n^{n}, n^{!}
Solution. Set A has n distinct elements. Then to define a function from A to A, we need to associate each element of set A to any one the n elements of set A. So total number of functions from set A to set A is equal to the number of ways of doing n jobs where each job can be done in n ways. The total number such ways is n × n × n × .... × n (n  times).
Hence the total number of functions from A to A is n^{n}.
Now for an onto function from A to A, we need to associate each element of A to one and only one element of A. So total number of onto functions from set A to A is equal to number of ways of arranging n elements in range (set A) keeping n elements fixed in domain (set A). n elements can be arranged in n! ways.
Hence, the total number of functions from A to A is n!.
Q. 5. If , then domain of f(x) is .... and its range is .................
Ans. (2, 1), [1, 1]
Solution. The given function is,
Combining these two inequalities, we get x ∈ (– 2, 1)
∴ Domain of f is (– 2, 1)
Also sin q always lies in [– 1, 1].
∴ Range of f is [– 1, 1]
Q. 6. There are exactly two distinct linear functions, ................., and ........... which map [– 1, 1] onto [0, 2].
Ans. x + 1 and  x+ 1
Solution. KEY CONCEPT : Every linear function is either strictly increasing or strictly decreasing. If f (x) = ax + b, Df = [p, q], Rf = [m, n]
Then f (p) = m and f (q) = n, if f (x) is strictly increasing and f (p) = n, f (q) = m, If f (x) is strictly decreasing function.
Let f (x) = ax + b be the linear function which maps [–1, 1] onto [0, 2]. then f (–1) = 0 and f (1) = 2 or f (–1) = 2 and f (1) = 0
Depending upon f (x) is increasing or decreasing respectively.
⇒ – a + b = 0 and a + b = 2 ....(1)
or – a + b = 2 and a + b = 0 ....(2)
Solving (1), we get a = 1, b = 1.
Solving (2), we get a = – 1, b = 1
Thus there are only two functions i.e., x + 1 and – x + 1.
Q. 7. If f is an even function defined on the interval (5, 5), then four real values of x satisfying the equation
are ................., ................., ................., and ...........
Ans.
Solution. Given that and f is an even function
Q. 8. If f(x) = sin^{2} x + then (gof) (x) = .................
Ans. 1
Solution.
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